A multiplicative process for generating a beta-like survival function with application to the UK EU referendum results

Trevor Fenner, Eric Kaufmann, Mark Levene, George Loizou

Human dynamics and sociophysics suggest statistical models that may explain and pro-vide us with better insight into social phenomena. Contextual and selection effects tend to produce extreme values in the tails of rank-ordered distributions of both census data and district-level election outcomes. Models that account for this nonlinearity generally outperform linear models. Fitting nonlinear functions based on rank-ordering census and election data can therefore improve the fit of aggregate voting models. This may help improve ecolo gical inference, as well as election forecasting in majoritarian systems.

We propose a generative multiplicative decrease model that gives rise to a rank-orderdistribution, and facilitates the analysis of the recent UK EU referendum results. We supply empirical evidence that the beta-like survival function, which can be generated directly from our model, is a close fit to the referendum results, and also may have predictive value when covariate data are available.

To obtain a rank ordering, assume that there are s voting districts, and that f (i, s) represents the expected proportion of the popular vote in the ith district, where the districts are ordered in descending order of their vote. We specify a multiplicative process, where μ(i, s), called the attrition function, is the probability that a potential vote is lost in the ith district. From this process we obtain the well-know renewal equation, whose solution is given by

f (i, s) =exp(-∫0i μ (i-t,s-t)dt) (1)

Moreover, when μ(i, s) is a mixture of preferential and uniform attrition, we can derive a closed form for f (i, s), given by

f (i, s) = C [0.5/(i+0.5)]α(1-i/s)β (2)

which we call the beta-like survival function as it can be viewed as a discrete version of the beta distribution, noting that the constant 0.5 is to prevent the first term being undefined when i = 0.

In our analysis of the per district Remain and Leave votes of the 2016 UK EU referendum, we demonstrate very good fits to the beta-like survival function for both Remain and Leave. We noticed that the power-law exponent α for Leave is significantly lower than that for Remain, while the decay exponent β for Leave is somewhat higher than that for Remain. This may indicate that the proportions of votes for Leave were more “stable” across thecountry than those for Remain. In other words, it is feasible that positive feedbacks driven by contextual effects on individual vote choice mattered more in Remain than Leave areas.

We also analysed, with the beta-like survival function, four census covariates, which theliterature suggests are associated with the Leave vote. The R2 values of the beta-like survival fits are much higher than those obtained from traditional linear regression, indicating that our methodology using beta-like survival functions may yield better predictive models than traditional ones based on linear regression of the raw covariate data.